A solid cone of height 8 cm and base radius 6 cm is melted and recast into identical cones, each of height 2 cm and diameter 1 cm. Find the number of cones formed.

Answer

To solve the problem, we will follow these steps:

### Step 1: Calculate the volume of the original solid cone.
The formula for the volume \( V \) of a cone is given by:
\[ V = \frac{1}{3} \pi r^2 h \]
where \( r \) is the radius and \( h \) is the height.

Given:
- Height \( h = 8 \) cm
- Base radius \( r = 6 \) cm

Substituting these values into the formula:
\[ V = \frac{1}{3} \pi (6)^2 (8) \]
Calculating \( (6)^2 = 36 \):
\[ V = \frac{1}{3} \pi (36)(8) = \frac{1}{3} \pi (288) = 96\pi \text{ cm}^3 \]

### Step 2: Calculate the volume of one identical cone.
The height of the identical cone is \( 2 \) cm, and the diameter is \( 1 \) cm, which gives a radius of:
\[ r = \frac{1}{2} \text{ cm} \]

Using the volume formula for the identical cone:
\[ V_{\text{identical}} = \frac{1}{3} \pi r^2 h \]
Substituting the values:
\[ V_{\text{identical}} = \frac{1}{3} \pi \left(\frac{1}{2}\right)^2 (2) \]
Calculating \( \left(\frac{1}{2}\right)^2 = \frac{1}{4} \):
\[ V_{\text{identical}} = \frac{1}{3} \pi \left(\frac{1}{4}\right)(2) = \frac{1}{3} \pi \left(\frac{2}{4}\right) = \frac{1}{3} \pi \left(\frac{1}{2}\right) = \frac{1}{6} \pi \text{ cm}^3 \]

### Step 3: Set up the equation to find the number of identical cones.
Let \( n \) be the number of identical cones formed. The volume of the original cone is equal to the total volume of the identical cones:
\[ n \times V_{\text{identical}} = V \]
Substituting the volumes we calculated:
\[ n \times \frac{1}{6} \pi = 96 \pi \]

### Step 4: Solve for \( n \).
Dividing both sides by \( \pi \):
\[ n \times \frac{1}{6} = 96 \]
Multiplying both sides by \( 6 \):
\[ n = 96 \times 6 = 576 \]

### Final Answer:
The number of identical cones formed is \( \boxed{576} \).

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